NCERT Solutions for Class 5th Maths Chapter 6 Shape puzzles, butter milk pouches, Which number am I?, Make your own numbers — Let Us Think
Book page 70–71 Updated on2026-09-19
Q1.
The given shapes stand for numbers between 1 and 24. The same shape denotes the same number across all problems. Find the numbers hiding in all the shapes.
The six shape problems printed on page 70. The same shape always stands for the same number.
Answer
One set of numbers that works: green pentagon = 1, orange diamond = 2, pink circle = 3, blue rectangle = 4, yellow parallelogram = 6, purple triangle = 12, white shape = 16, blue half-circle = 18.
Here is how to unlock the puzzle, step by step. Start with the clue that gives away a number all by itself.
Look at (d): diamond × pentagon = diamond. A number stays the same only when you multiply it by 1. So the green pentagon = 1.
Look at (b): rectangle × rectangle = white shape. The rectangle times itself must stay below 24. So the rectangle can only be 1, 2, 3 or 4. Take the biggest, rectangle = 4. Then white shape = 4 × 4 = 16.
Look at (a): rectangle × circle = triangle. Try circle = 3. Then triangle = 4 × 3 = 12.
Look at (c): parallelogram × diamond = triangle. We need two numbers whose product is 12. Take parallelogram = 6 and diamond = 2. Check (d) again: 2 × 1 = 2 ✓.
Look at (e): parallelogram × circle = half-circle. 6 × 3 = 18. That is under 24, so it is allowed.
Check (f): half-circle × rectangle = 18 × 4 = 72, and parallelogram × triangle = 6 × 12 = 72. Both sides give 72 ✓.
Shape
Number
Check
Green pentagon
1
(d) 2 × 1 = 2
Orange diamond
2
(c) 6 × 2 = 12
Pink circle
3
(a) 4 × 3 = 12
Blue rectangle
4
(b) 4 × 4 = 16
Yellow parallelogram
6
(e) 6 × 3 = 18
Purple triangle
12
(f) 6 × 12 = 72
White shape
16
(b) 4 × 4 = 16
Blue half-circle
18
(f) 18 × 4 = 72
Why it happens: Clue (d) is the key that opens the door. Multiplying by 1 never changes a number, so the pentagon had to be 1. After that, every other clue only had to agree with the ones before it.
Did you know? This puzzle has more than one answer. If you take the rectangle as 3 instead of 4, other sets of numbers also fit. Try it and see what you get — just make sure every shape stays between 1 and 24.
Q2.
Place the digits 2, 5, and 3 appropriately to get a product close to 100. Share your reasoning in class.
Answer
Put 53 in the two-digit box and 2 in the one-digit box: 53 × 2 = 106. That is the closest you can get to 100.
There are only six ways to arrange the three digits. Try them all and compare.
Sample reasoning to share in class: I want an answer near 100. If the one-digit number is big, like 5, the answer shoots past 100. So I kept the small digit 2 as the multiplier. Then I made the two-digit number as close to 50 as I could, because 50 × 2 is exactly 100. The biggest number I could build from 5 and 3 is 53, and 53 × 2 = 106. Only 6 away.
Q3.
A dairy has packed butter milk pouches in the following manner. Find the number of pouches kept in each arrangement. One is done for you. (First arrangement: 30 × 2 = 60.)
Answer
All four arrangements hold 60 pouches. Only the way they are grouped changes.
First arrangement (already done): pouches sit in pairs. There are 6 pairs in each row and 5 rows, so 6 × 5 = 30 pairs. That is 30 × 2 = 60.
Second arrangement: pouches sit in threes. There are 4 threes in each row and 5 rows, so 4 × 5 = 20 groups. That is 20 × 3 = 60.
Third arrangement: tall blocks of 2 across and 5 down, so 10 pouches in each block. There are 6 blocks. That is 6 × 10 = 60.
Fourth arrangement: long rows of 12 pouches, and there are 5 rows. That is 5 × 12 = 60.
The pouches never change. Only the size of the group changes, so the two numbers swap around.
30 × 2 = 60 20 × 3 = 60 6 × 10 = 60 5 × 12 = 60
Why it happens: When you make the groups bigger, you need fewer of them. Twice the group size means half the number of groups, so the total stays exactly the same.
Q4.
What other groups can you make? (with the butter milk pouches)
Answer
You can make twelve different groupings of 60 pouches. Every pair of numbers that multiplies to 60 gives one.
Number of groups
Group size
Total
1
60
60
2
30
60
3
20
60
4
15
60
5
12
60
6
10
60
10
6
60
12
5
60
15
4
60
20
3
60
30
2
60
60
1
60
How to find them all without missing any:
Start at 1. Ask: does 60 share out evenly into 1 group? Yes — one group of 60.
Try 2, then 3, then 4 and so on. Each time, ask whether 60 divides evenly.
Write the partner next to it. 60 ÷ 3 = 20, so 3 and 20 are partners.
Stop when the pairs start repeating. After 6 × 10 you meet 10 × 6, which you already have.
Check it yourself: 7 does not work, because 60 pouches cannot be shared into groups of 7 without some left over. Nor do 8, 9, 11, 13 or 14. That is why they are missing from the table.
Q5.
Which number am I? I am a two-digit number. Find me with the help of the following clues. (a) I am greater than 8. (b) I am not a multiple of 4. (c) I am a multiple of 9. (d) I am an odd number. (e) I am not a multiple of 11. (f) I am less than 50. (g) My ones digit is even. (h) My tens digit is odd.
Answer
The number is 18.
Take the clues one at a time and cross out numbers as you go.
Clue (c) — I am a multiple of 9. The two-digit multiples of 9 are 18, 27, 36, 45, 54, 63, 72, 81, 90, 99.
Clue (f) — I am less than 50. Only 18, 27, 36, 45 are left.
Clue (g) — my ones digit is even. 27 ends in 7 and 45 ends in 5, both odd. Cross them out. Left: 18, 36.
Clue (b) — I am not a multiple of 4. 36 = 4 × 9, so cross out 36. Left: 18.
Clue (h) — my tens digit is odd. 18 has 1 in the tens place, and 1 is odd ✓.
Multiples of 9 below 50: 18, 27, 36, 45 Ones digit even → 18, 36 Not a multiple of 4 → 18
One clue does not fit: Clue (d) says the number is odd, but 18 is even. No number can have an even ones digit (clue g) and still be odd (clue d) — those two clues fight each other. Every other clue points firmly to 18, so 18 is the answer. If your teacher asks, say so honestly: clue (d) looks like a printing slip.
Q6.
Did you use all the clues to find the number? Which clues did not help you in finding the number?
Answer
No — two clues were of no help at all: (a) and (e).
Clue (a) — I am greater than 8. The puzzle already told us the number has two digits. Every two-digit number is 10 or more, so it is already bigger than 8. This clue removes nothing.
Clue (e) — I am not a multiple of 11. The only two-digit multiples of 11 are 11, 22, 33, 44, 55, 66, 77, 88 and 99. None of them is a multiple of 9 except 99, and 99 was already gone because it is bigger than 50. So this clue removes nothing either.
The clues that did the work were (c) multiple of 9, (f) less than 50, (g) ones digit even, and (b) not a multiple of 4.
Why it happens: A clue only helps if it throws some numbers away. If everything left over already obeys the clue, the clue is just extra words. Good puzzle-solvers always check which clues are actually doing the work.
Q7.
Make your own numbers. Choose any two numbers and one operation from the grid. Try to make all the numbers between 0 and 20. For example, 2 can be formed as 4 – 2. Could you make all the numbers?
The number grid and the four operation cards printed on page 71.
Answer
You can make every number from 1 to 20 except 19. The number 0 cannot be made either. So 19 numbers out of 21 are possible.
Here is one way to make each one. There are often several.
Number
One way
Number
One way
0
not possible
11
36 − 25
1
3 − 2
12
4 × 3
2
4 − 2
13
3 + 10
3
5 − 2
14
4 + 10
4
12 ÷ 3
15
5 + 10
5
3 + 2
16
4 + 12
6
2 × 3
17
5 + 12
7
5 + 2
18
36 ÷ 2
8
4 × 2
19
not possible
9
36 ÷ 4
20
5 × 4
10
12 − 2
Why 0 is impossible: To get 0 by subtracting you need two equal numbers, and every number in the grid is different. To get 0 by multiplying you need a 0 in the grid, and there is none. Adding and dividing two positive numbers can never give 0 either.
Try this: Make a list of every answer you can reach with just the small numbers 2, 3, 4, 5, 10 and 12. You will find most of 0 to 20 hiding there.
Q8.
Which numbers could you not make? Is it possible to make these numbers using three numbers? You can use two operations, if needed.
Answer
The numbers you could not make with two grid numbers are 0 and 19. With three numbers and two operations, both become easy.
19 = 25 − 4 − 2, because 25 − 4 = 21 and 21 − 2 = 19.
19 = 36 ÷ 2 + ... no, that gives 18 and there is no 1 in the grid. So this one does not work — always check.
Why the third number helps: With two numbers you only get one jump. With three numbers you get two jumps, so you can overshoot and then come back. That is exactly what 12 + 10 − 3 does: it jumps to 22 and steps back to 19.
Q9.
Which numbers between 0–20 can you get in more than one way?
Answer
Most of them. Only 11, 16, 17 and 18 have just one way each. Every other number from 1 to 20 can be made in two or more ways.
Number
Ways to make it
1
3 − 2, 5 − 4, 4 − 3
2
4 − 2, 5 − 3, 12 − 10, 10 ÷ 5, 4 ÷ 2
3
5 − 2, 36 ÷ 12
4
12 ÷ 3, 100 ÷ 25
5
3 + 2, 25 ÷ 5, 10 − 5, 10 ÷ 2
6
4 + 2, 2 × 3, 10 − 4, 12 ÷ 2
7
5 + 2, 3 + 4, 12 − 5, 10 − 3
8
5 + 3, 10 − 2, 4 × 2, 12 − 4
9
12 − 3, 36 ÷ 4
10
12 − 2, 100 ÷ 10, 5 × 2
11
36 − 25 only
12
2 + 10, 4 × 3, 36 ÷ 3
13
3 + 10, 25 − 12
14
2 + 12, 4 + 10
15
5 + 10, 3 + 12, 25 − 10, 5 × 3
16
4 + 12 only
17
5 + 12 only
18
36 ÷ 2 only
20
5 × 4, 25 − 5, 100 ÷ 5, 2 × 10
Check it yourself: The small numbers 2, 3, 4, 5, 10 and 12 pair up in many ways, so the small answers have many routes. The big numbers 25, 36 and 100 only fit a few places, so 11, 16, 17 and 18 have just one route each.